Depends a lot what the original image is (size and content).
If this is mostly a photo, then you have to use JPEG to get the file small, and file size will be a compromise between image size and quality (hint: in Gimp's advanced options, there is a "chroma subsampling" option which is what gives the more efficient size reduction for the smallest quality loss, so start there).
If it's CGI (text, logo) with large uniform areas, then PNG may be more efficient than JPEG. You can try to produce an indexed PNG (in Gimp: Image>Mode>Indexed before exporting to PNG).
Gimp has a "Save for web" plugin that lets you see what the final image looks like and how big the file is.